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Theorem iooval2 13294

Description: Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

**Assertion**
Ref	Expression
iooval2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem iooval2**
Step	Hyp	Ref	Expression
1		iooval 13285	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
2		elioore 13291	. . . . . 6 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
3	2	ssriv 3937	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
4	1, 3	eqsstrrdi 3979	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ)
5		dfss2 3919	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ ↔ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
6	4, 5	sylib 218	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
7		inrab2 4269	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
8		ressxr 11176	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
9		sseqin2 4175	. . . . . 6 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
10	8, 9	mpbi 230	. . . . 5 ⊢ (ℝ^* ∩ ℝ) = ℝ
11	10	rabeqi 3412	. . . 4 ⊢ {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
12	7, 11	eqtri 2759	. . 3 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
13	6, 12	eqtr3di 2786	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
14	1, 13	eqtrd 2771	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {crab 3399 ∩ cin 3900 ⊆ wss 3901 class class class wbr 5098 (class class class)co 7358 ℝcr 11025 ℝ^*cxr 11165 < clt 11166 (,)cioo 13261

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-pre-lttri 11100 ax-pre-lttrn 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-ioo 13265

This theorem is referenced by: elioo2 13302 ioomax 13338 ioopos 13340 dfioo2 13366

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